Young was the third overall pick in the 2006 draft for the Tennessee Titans but has not made a regular season roster since 2011 with the Philadelphia Eagles.
The former Texas QB had been trying out during the Browns optional workouts along with Tyler Thigpen, who the Browns are also said to be signing to a deal.
NFL.com's Ian Rapoport tweeted: "The #Browns are expected to sign QB Vince Young to a one-year deal after his tryout, source says. Very cool comeback story."
Rapoport also mentioned that Young had been offered a job at the University of Texas but he has instead opted for another shot at being an NFL quarterback.
The Browns now have three quarterbacks on their roster, with Brian Hoyer the presumptive starter at the moment. However, they should also be expected to draft an additional passer at some point during the draft next week.